Neanderthals weaned their babies between 9 and 18 months of age

Or at least one of them did. 

The finding is the product of detailed analysis of milk tooth formation in one infant Neanderthal from Scladina cave (Belgium) and comparison with many monkey teeth. The researchers concluded that the barium accumulation in the teeth correlates tightly with breastfeeding and gives information on this with almost a day of precision.

This Neanderthal kid was exclusive breastfed up to the age of nine months and then had another nine months of gradual weaning, eating also other foods, as well as its mother's milk. 

This is probably much more than the average breastfeeding in our modern societies but less than it has been documented among some hunter-gatherers like Bushmen, who may well partly breastfeed their children for up to four years, what acts as (unsafe) contraceptive. Chimpanzees seem to breastfeed their infants for some 5.3 years, while non-civilized humans (H. sapiens) have ranges of around 2.4 years instead.

Sources: Science Daily, Paleorama[es].

Ref. Christine Austin, Tanya M. Smith, Asa Bradman, Katie Hinde, Renaud Joannes-Boyau, David Bishop, Dominic J. Hare, Philip Doble, Brenda Eskenazi, Manish Arora. Barium distributions in teeth reveal early-life dietary transitions in primates. Nature, 2013; DOI: 10.1038/nature12169